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TTF Workshop
25-28 March, 2008
Boulder, CO
Parallel and Perpendicular Flowsdriven by Stochastic Magnetic Fields in MST
D.L. Brower
W.X. Ding, T. YatesUniversity of California at Los Angeles
G. Fiksel, J.S. Sarff, S.C. Prager and the MST GroupUniversity of Wisconsin-Madison
D. CraigWheaton College
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GOAL: Identify the role of stochastic magnetic fields in momentumtransport and flow generation during reconnection
Results:
(1) transport of parallel momentum from stochastic fields is measuredand found to be significant in the plasma core……matching parallel velocity relaxation
(2) Stochastic field driven charge transport produces localized perpendicular flow and radial electric field (shear)
…….to be presented at poster
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MST Reversed-Field Pinch (RFP) is toroidal configuration with relatively weak toroidal magnetic field BT ( i.e., BT ~ Bp)
Madison Symmetric Torus - MST
R0 = 1.5 m, a = 0.51 m, Ip < 600 kA BT ~ 3-4 kG, ne ~ 1019 m-3, Te0 < 1.3 keV τE ~ 10 ms, β =
/B2(a)=15%
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Field Line Puncture Plot
Toro
idal
Ang
le / π
r/a
Puncture Plot of B Field in MST
magnetic field in MST is typically stochastic andshould contribute to momentum transport
• J. Finn (1991?) suggests momentum diffusion with D ~ Dm cs• Plugging in MST parameters: τm ~ 1 ms in ambient case
τm ~ 100 µs during relaxation event
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• Momentum density in z direction being transferred in xdirection is
• Assuming a strong B field with small fluctuations and taking zalong B, we can write the contribution from parallel streamingparticles using vz ≅ v|| and vx ≅ v|| (bx/B) as
• Averaging over flux surface (z and y directions) we get theaverage momentum flux.
momentum transport by stochastic B field
vdvvmf xz3
! v)(x,
!
f (x,v)" m v||2bx
Bd3v =
p||bx
B
B
bp x >< ||
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• If = 0, then we are left with
• In order for the momentum at x to change, need more fluxcoming in from –x than that leaving toward +x.
• above relation provides a direct measure of the transport of momentum along the mean magnetic field direction due to stochastic magnetic field.
• Substituting , we get two terms
!
Flux : "rmom
=< ˜ p
||˜ b x >
B
!!
"
#
$$
%
& ><'(
)
>
B
&
' (
)
* + $ n%
< ˜ T ||
˜ b x
>
B
&
' (
)
* +
!
˜ p //
= T//
˜ n + n ˜ T //
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Momentum transported from core to the edgeduring reconnection
During reconnection, parallel momentum redistributes within ~100 µs,much faster than classical collision time.
25
20
15
10
km
/s
-2 -1 0 1 2time [ms]
Toroidal flow r/a=0
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Stochastic field-driven transport is directly measured(first term only)
• Measure T//,ion with Rutherford Scattering (or CHERS)• Measure br and B with FIR laser Faraday Rotation• Measure with differential FIR interferometer
• Measurement details provided at poster….!
˜ n and " ˜ n
!
"< #v||
>
"t~ $T
||%
< ˜ n ˜ b r
>
B
&
' (
)
* + $ n%
< ˜ T ||
˜ b r
>
B
&
' (
)
* +
Convection
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FIR Polarimeter-Interferometer System
MST!
" ~ ndl# + ˜ n dl#
Interferometer
11 chords, Δx~ 8 cm, Δφ~ 0.05o , Δt~ 1µs
!
" ˜ #
"x"x ~ 1mm
!
" ~ nr B • d
r l + n ˜
r
b • dr l + ˜ n
r B • d
r l ###
Differential Interferometer
Faraday rotation
!
˜ n
!
" ˜ n
!
˜ b and ˜ j
!
˜
r
b • dr l
L
" = µ0 ˜ I # $˜ %
1& ˜ %
2
cF n 0' ˜ j #
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At x=-2 cm
Momentum flux and parallel momentum change
!
"mom
= T//,i
< ˜ n ̃ b r
>
B
Magnetic fluctuation-induced (convective) flux drives parallel momentum drop!
r / a = 0.04 -T//,i"< ˜ n ̃ b
r>
B
# < $V// >
#t-8
-6
-4
-2
0
2
4
N/m
3
1.51.00.50.0-0.5-1.0Time [ms]
!
" < #V// >
"t+ T//,i$
< ˜ n ̃ b r
>
B% 0
m=1, n=6,7,8,9
800
600
400
200
0
Ti [e
V]
1.51.00.50.0-0.5-1.0Time [ms]
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Momentum flux changes direction away from magnetic axis
!
r /a = 0.04 " < #V
//>
"t
-$ •%mom
!
r /a = 0.45
-8
-6
-4
-2
0
2
4
N/m
3
1.51.00.50.0-0.5-1.0Time [ms]
4
2
0
-2
-4
N/m
3
1.51.00.50.0-0.5-1.0time [ms]
Direction change consistent with momentum transport(not total momentum loss)
30
20
10
0
-10
[km
/s]
1.51.00.50.0-0.5-1.0Time [ms]
40
30
20
10
0
-10
V// [
km
/s]
1.51.00.50.0-0.5-1.0Time [ms]
!
< "V//
>
# $ •%mom& dt
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Summary: Stochastic magnetic fields drive parallel and perpendicular flows
Measurements indicate :
Momentum transport(convective)
!
"rmom
=< ˜ p //
˜ b r >stochastic magnetic field
tearing mode coupling
Parallel Velocity Relaxation
Charge Transport
Perpendicular Zonal Flow
Radial Electric Field !
Er
= "Te#n
e
nee
Increasing Density Gradient
!
"q = "ri#"r
e
!
V"i
= #Er
B